This comes with having never done it... So someone feel free to jump in and correct me if I'm wrong.

If you let someone adopt your trackable, it is no longer your trackable. It is now theirs. It will no longer appear in your owned trackable list, but you could certainly discover it and have it appear on your discovered trackable list. It will appear on the owned trackable list for the person who received the adoption.

The icon follows the person that owns it. The person that gave it up doesn't have the icon anymore - the person that gained it gets the icon. For purposes of the icon, it's as if the trackable was originally registered to the person who gained it._________________Hmm...